Mathematical Symbols

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List of mathematical symbols

This is a list of symbols used in all branches ofmathematics to express a formula or to represent aconstant .

A mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations, a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. However, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.

Each symbol is shown both inHTML , whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image usingTeX .

Contents

Guide Basic symbols Symbols based on equality Symbols that point left or right Brackets Other non-letter symbols Letter-based symbols Letter modifiers Symbols based on Latin letters Symbols based on Hebrew or Greek letters Variations See also References External links

Guide

This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. For a related list organized by mathematical topic, see List of mathematical symbols by subject. That list also includes LaTeX and HTML markup, and Unicode code points for each symbol (note that this article doesn't have the latter two, but they could certainly be added).

There is a Wikibooks guide for using maths in LaTeX,[1] and a comprehensive LaTeX symbol list.[2] It is also possible to check to see if a Unicode code point is available as a LaTeX command, or vice versa.[3] Also note that where there is no LaTeX command natively available for a particular symbol (although there may be options that require adding packages), the symbol could be added via other options, such as setting the document up to support Unicode,[4] and entering the character in a variety of ways (e.g. copying and pasting, keyboard shortcuts, the \unicode{} command[5]) as well as other options[6] and extensive additional information.[7][8]

Basic symbols: Symbols widely used in mathematics, roughly through first-year calculus. More advanced meanings are included with some symbols listed here. Symbols based on equality "=": Symbols derived from or similar to the equal sign, including double-headed arrows. Not surprisingly these symbols are often associated with an equivalence relation. Symbols that point left or right: Symbols, such as < and >, that appear to point to one side or another. Brackets: Symbols that are placed on either side of a variable or expression, such as|x |. Other non-letter symbols: Symbols that do not fall in any of the other categories. Letter-based symbols: Many mathematical symbols are based on, or closely resemble, a letter in some alphabet. This section includes such symbols, including symbols that resemble upside-down letters. Many letters have conventional meanings in various branches of mathematics and physics. These are not listed here. TheSee also section, below, has several lists of such usages.

Letter modifiers: Symbols that can be placed on or next to any letter to modify the letter's meaning. Symbols based on Latin letters, including those symbols that resemble or contain anX .δ, Δ, π, Π, σ, Σ, Φ. Note: symbols resembling Λ are grouped with "V" under Latin letters ,ב ,א .Symbols based on Hebrew or Greek letters e.g Variations: Usage in languages written right-to-left

Basic symbols Name Symbol Symbol Read as Explanation Examples in HTML in TeX Category addition plus; means the sum of and . add 4 + 6 4 6 2 + 7 = 9 arithmetic + disjoint union the disjoint A1 + A2 means the disjoint union of A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒ union of ... sets and . and ... A1 A2 A1 + A2 = {(3, 1), (4, 1), (5, 1), (6, 1), (7, 2), (8, 2), (9, 2), (10, 2)} set theory subtraction minus; 36 − 11 means the subtraction of11 take; 36 − 11 = 25 from . subtract 36 arithmetic negative sign negative; minus; −3 means the additive inverse of the −(−5) = 5 − the opposite number 3. of arithmetic set-theoretic A − B means the set that contains all complement the elements of A that are not in B. minus; {1, 2, 4} − {1, 3, 4} = {2} without (∖ can also be used for set-theoretic set theory complement as described below.) plus-minus plus or minus 6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. arithmetic ± \pm plus-minus 10 ± 2 or equivalently 10 ± 20% plus or minus means the range from 10 − 2 to If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm. measurement 10 + 2. minus-plus 6 ± (3 ∓ 5) means 6 + (3 − 5) and minus or plus cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y). ∓ \mp 6 − (3 + 5). arithmetic multiplication times; 3 × 4 or 3 ⋅ 4 means the multiplication 7 ⋅ 8 = 56 multiplied by of 3 by 4. arithmetic dot product scalar product dot u ⋅ v means the dot product ofvectors (1, 2, 5) ⋅ (3, 4, −1) = 6 linear algebra u and v × vector

algebra \times ⋅ cross product vector \cdot product i j k u × v means the cross product of cross (1, 2, 5) × (3, 4, −1) = 1 2 5 = (−22, 16, −2) · vectors u and v linear algebra 3 4 −1 vector algebra placeholder A · means a placeholder for an argument of a function. Indicates the (silent) functional nature of an expression | · | functional without assigning a specific symbol for analysis an argument. division (Obelus) 2 ÷ 4 = 0.5 6 ÷ 3 or 6 ⁄ 3 means the division of 6 divided by; by 3. over 12 ⁄ 4 = 3 arithmetic ÷ \div quotient group G / H means the quotient of groupG {0, a, 2a, b, b + a, b + 2a} / {0, b} = {{0, b}, {a, b + a}, {2a, b + 2a}} ⁄ mod modulo its subgroup H. group theory quotient set A/~ means the set of all ~ equivalence If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then mod classes in A. ℝ/~ = {x + n : n ∈ ℤ, x ∈ [0,1)}. set theory square root (radical symbol) √x means the nonnegative number √4 = 2 the (principal) whose square is x. square root of

\surd real numbers √ complex \sqrt{x} square root If z = r exp(iφ) is represented in polar the (complex) coordinates with , then square root of −π < φ ≤ π √−1 = i √z = √r exp(iφ/2). complex numbers summation ∑ \sum sum over ... from ... to ... means . of calculus indefinite integral or antiderivative indefinite

integral of ∫ f(x) dx means a function whose - OR - derivative is f. the antiderivative of calculus definite integral b ∫ integral from ∫ f(x) dx means the signed area 3 3 \int ... to ... of ... a b x2 dx = b − a between the x-axis and the graph of the ∫ a 3 with respect function f between x = a and x = b. to calculus

f ds means the integral of f along line integral ∫ C b line/ path/ the curve C, ∫ f(r(t)) |r'(t)| dt, where curve/ integral a r is a parametrization of C. (If the curve of ... along ...

calculus is closed, the symbol ∮ may be used instead, as described below.) Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the

Contour symbol ∯ would be more appropriate. integral; A third related symbol is the closed closed line volume integral, denoted by the symbol integral 1 If is a Jordan curve about 0, then . ∰ . C ∮ C z dz = 2πi ∮ \oint contour integral of The contour integral can also frequently calculus be found with a subscript capital letter

C, ∮ C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's

Law, a subscript capital S, ∮ S, is used to denote that the integration is over a closed surface.

… \ldots

⋯ \cdots ellipsis

⋮ and so forth Indicates omitted values from a pattern. 1/2 + 1/4 + 1/8 + 1/16 + ⋯ = 1 everywhere \vdots ⋰

⋱ \ddots therefore therefore; Sometimes used in proofs before so; All humans are mortal. Socrates is a human.∴ Socrates is mortal. ∴ \therefore logical consequences. hence everywhere because because; Sometimes used in proofs before 11 is prime ∵ it has no positive integer factors other than itself and one. ∵ \because since reasoning. everywhere ! factorial factorial means the product . combinatorics logical The statement !A is true if and only if A !(!A) ⇔ A negation is false. x ≠ y ⇔ !(x = y) not A slash placed through another propositional operator is the same as "!" placed in logic front.

(The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.) The statement ¬A is true if and only if A is false.

logical A slash placed through another

¬ negation operator is the same as "¬" placed in \neg ¬(¬A) ⇔ A not front. x ≠ y ⇔ ¬(x = y) ˜ propositional logic (The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use ! but this is avoided in mathematical texts.) proportionality is proportional y ∝ x means that y = kx for some to; if y = 2x, then y ∝ x. ∝ \propto constant k. varies as everywhere infinity ∞ is an element of the extended

infinity number line that is greater than all real ∞ \infty numbers numbers; it often occurs inlimits . ■

□ \blacksquare end of proof QED; Used to mark the end of a proof. tombstone; ∎ \Box Halmos (May also be written Q.E.D.) finality symbol everywhere ▮ \blacktriangleright

‣

Symbols based on equality Name Symbol Symbol Read as Explanation Examples in HTML in TeX Category equality

is equal to; means and represent the same thing or value. = equals everywhere inequality means that and do not represent the same thing or value. is not equal to; ≠ \ne does not equal (The forms !=, /= or <> are generally used in programming everywhere languages where ease of typing and use ofASCII text is preferred.) approximately equal x ≈ y means x is approximately equal toy . is approximately π ≈ 3.14159 equal to This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒.

everywhere ≈ \approx isomorphism G ≈ H means that group G is isomorphic (structurally identical) to group H. is isomorphic to Q8 / C2 ≈ V group theory (≅ can also be used for isomorphic, as described below.) probability distribution X ~ D, means the random variable X has the probability distribution X ~ N(0,1), the standard has distribution D. normal distribution statistics row equivalence is row equivalent A ~ B means that B can be generated by using a series of to elementary row operations on A matrix theory same order of magnitude m ~ n means the quantities m and n have the same order of 2 ~ 5 roughly similar; magnitude, or general size. poorly 8 × 9 ~ 100 approximates; (Note that ~ is used for an approximation that is poor, otherwise use is on the order of ≈ .) but π2 ≈ 10 approximation ~ \sim theory similarity [9] △ABC ~ △DEF means triangle ABC is similar to (has the same is similar to shape) triangle DEF. geometry asymptotically equivalent is asymptotically f ~ g means . x ~ x+1 equivalent to asymptotic analysis equivalence relation 1 ~ 5 mod 4 are in the same a ~ b means (and equivalently ). equivalence class everywhere =:

\equiv ≡ definition x := y, y =: x or x ≡ y means x is defined to be another name fory , under certain assumptions taken in context. :\Leftrightarrow is defined as; is equal by :⇔ (Some writers use ≡ to mean congruence). definition to \triangleq everywhere P Q means P is defined to be logically equivalent to Q. ≜ ⇔

\overset{\underset{\mathrm{def}} ≝ {}}{=}

≐ \doteq congruence △ABC ≅ △DEF means triangle ABC is congruent to (has the same is congruent to measurements as) triangle DEF. geometry

≅ \cong isomorphic G ≅ H means that group G is isomorphic (structurally identical) to group H. is isomorphic to V ≅ C2 × C2 abstract algebra (≈ can also be used for isomorphic, as described above.) congruence relation

... is congruent to a ≡ b (mod n) means a − b is divisible by n 5 ≡ 2 (mod 3) ≡ \equiv ... modulo ... modular arithmetic material A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y + 2 ⇔ x + 3 = y ⇔ \Leftrightarrow equivalence if and only if; ↔ \iff iff propositional logic

\leftrightarrow

:= Assignment Let a := 3, then... is defined to be A := b means A is defined to have the valueb . f(x) := x + 3 =: everywhere

Symbols that point left or right Name Symbol Symbol Read as Explanation Examples in HTML in TeX Category strict inequality is less than, means x is less than y. is greater than means x is greater than y. < order theory > proper subgroup is a proper means is a proper subgroup of . subgroup of H G group theory significant (strict) inequality is much less x ≪ y means x is much less than y. than, 0.003 ≪ 1000000 means x is much greater than y. is much greater x ≫ y than order theory asymptotic comparison

is of smaller ≪ f ≪ g means the growth of f is asymptotically bounded byg . order than, \ll (This is I. M. Vinogradov's notation. Another notation is theBig O notation, x is of greater x ≪ e ≫ \gg which looks like .) order than f = O(g) analytic number theory absolute continuity is absolutely means that is absolutely continuous with respect to , i.e., If is the counting measure on and is continuous with whenever , we have . the Lebesgue measure, then . respect to measure theory

inequality x ≤ y means x is less than or equal to y. means x is greater than or equal toy . is less than or x ≥ y (The forms <= and >= are generally used in programming languages, equal to, 3 ≤ 4 and 5 ≤ 5 where ease of typing and use ofASCII text is preferred.) is greater than 5 ≥ 4 and 5 ≥ 5 or equal to (≦ and ≧ are also used by some writers to mean the same thing as≤ and order theory ≥, but this usage seems to be less common.) subgroup Z ≤ Z ≤ is a subgroup of H ≤ G means H is a subgroup of G. A3 ≤ S3 \le group theory ≥ \ge If reduction is reducible to A ≤ B means the problem A can be reduced to the problemB . Subscripts computational can be added to the ≤ to indicate what kind of reduction. then complexity theory

congruence relation ... is less than ... is greater than 10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10 ... modular ≦ arithmetic \leqq ≧ vector inequality x ≦ y means that each component of vectorx is less than or equal to each \geqq corresponding component of vectory . ... is less than or means that each component of vectorx is greater than or equal to equal... is x ≧ y each corresponding component of vectory . greater than or equal... It is important to note that x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also order theory true. Karp reduction is Karp reducible to; is polynomial- [10] time many-one L1 ≺ L2 means that the problem L1 is Karp reducible to L2. If L1 ≺ L2 and L2 ∈ P, then L1 ∈ P. reducible to computational ≺ complexity \prec theory ≻ \succ Nondominated order is nondominated means that the element P is nondominated by elementQ .[11] If P ≺ Q then by P ≺ Q 1 2 Multi-objective optimization normal ◅ subgroup ▻ \triangleleft is a normal N ◅ G means that N is a normal subgroup of groupG . Z(G) ◅ G ◅ \triangleright subgroup of group theory ▻ ideal is an ideal of I ◅ R means that I is an ideal of ring R. (2) ◅ Z ring theory antijoin R ▻ S means the antijoin of the relationsR and S, the tuples in R for which the antijoin of there is not a tuple in S that is equal on their common attribute names. relational algebra material implication A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. ⇒ implies; 2 is true, but (→ may mean the same as ⇒, or it may have the meaning forfunctions x = 6 ⇒ x − 5 = 36 − 5 = 31 if ... then 2 → \Rightarrow given below.) x − 5 = 36 −5 = 31 ⇒ x = 6 is in general ⊃ \rightarrow propositional (⊃ may mean the same as ⇒,[12] or it may have the meaning forsuperset false (since x could be −6). \supset logic, Heyting given below.) algebra

subset (subset) means every element ofA is also an element of B.[13] (A ∩ B) ⊆ A A ⊆ B ⊆ is a subset of \subseteq (proper subset) A ⊂ B means A ⊆ B but A ≠ B. ℕ ⊂ ℚ ⊂ \subset set theory (Some writers use the symbol ⊂ as if it were the same as ⊆.) ℚ ⊂ ℝ

superset A ⊇ B means every element ofB is also an element of A. ⊇ (A ∪ B) ⊇ B is a superset of A ⊃ B means A ⊇ B but A ≠ B. \supseteq (Some writers use the symbol ⊃ as if it were the same as ⊇.) ℝ ⊃ ℚ ⊃ \supset set theory compact embedding \Subset is compactly A ⋐ B means the closure of B is a compact subset of A. contained in set theory function arrow from ... to → f: X → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ ∪ {0} be defined by f(x) := x2. \to set theory, type theory function arrow

maps to means the function f maps the element a to the element b. Let (the successor function). ↦ \mapsto f: a ↦ b f: x ↦ x + 1 set theory Converse a ← b means that for the propositionsa and b, if b implies a, then a is the implication converse implication of b.a to the element b. This reads as "a if b", or "not b ← \leftarrow .. if .. without a". It is not to be confused with theassignment operator in logic computer science. subtype If and then is a subtype of T <: T means that T is a subtype of T . S <: T T <: U S <: U 1 2 1 2 (transitivity). <: type theory cover <· {1, 8} <• {1, 3, 8} among the subsets of is covered by x <• y means that x is covered by y. {1, 2, ..., 10} ordered by containment. order theory entailment A B means the sentence A entails the sentence B, that is in every model entails ⊧ A ⊧ A ∨ ¬A ⊧ \vDash in which A is true, B is also true. model theory inference infers; is derived from x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A propositional ⊢ logic, predicate \vdash logic partition is a partition of p ⊢ n means that p is a partition of n. (4,3,1,1) ⊢ 9, number theory bra vector the bra ...; ⟨φ| means the dual of the vector φ| ⟩, a linear functional which maps a ket | ⟨| \langle the dual of ... ψ⟩ onto the inner product ⟨φ|ψ⟩. Dirac notation ket vector A qubit's state can be represented asα |0⟩+ the ket ...; |φ⟩ means the vector with labelφ , which is in a Hilbert space. β|1⟩, where α and β are complex numbers s.t. |⟩ \rangle the vector ... |α|2 + |β|2 = 1. Dirac notation

Brackets Name Symbol Symbol Read as Explanation Examples in HTML in TeX Category

combination; means (in the case of n = positive integer) the number of binomial coefficient combinations of k elements drawn from a set ofn elements. {\ \choose\ n choose k n } (This may also be written as C(n, k), C(n; k), nCk, Ck, or combinatorics .)

multiset coefficient \left(\!\!{\ u multichoose \choose\ k (when u is positive integer) }\!\!\right) combinatorics means reverse or rising binomial coefficient.

absolute |3| = 3 value; modulus |–5| = |5| = 5 |x| means the distance along thereal line (or across the absolute complex plane) between x and zero. value of; | i | = 1 modulus of numbers | 3 + 4i | = 5 Euclidean norm or Euclidean length or For x = (3,−4) |x| means the (Euclidean) length ofvector x. magnitude |...| | \ldots | Euclidean \!\, norm of geometry determinant determinant |A| means the determinant of the matrixA of matrix theory cardinality cardinality of; |X| means the cardinality of the setX . size of; |{3, 5, 7, 9}| = 4. order of (# may be used instead as described below.) set theory norm norm of; ‖ x ‖ means the norm of the element x of a normed vector ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ length of space.[14] linear algebra

‖...‖ \| \ldots \| nearest \!\, integer function ‖x‖ means the nearest integer tox . ‖1‖ = 1, ‖1.6‖ = 2, ‖−2.4‖ = −2, ‖3.49‖ = 3 nearest (This may also be written [x], ⌊x⌉, nint(x) or Round(x).) integer to numbers set brackets { , } {\{\ ,\!\ the set of ... {a,b,c} means the set consisting ofa , b, and c.[15] ℕ = { 1, 2, 3, ... } \}} \!\, set theory

\{\ :\ \} { : } \!\, set builder notation {x : P(x)} means the set of all x for which P(x) is true.[15] {x | \{\ |\ \} the set of ... {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4 } { | } P(x)} is the same as {x : P(x)}. \!\, such that set theory { ; } \{\ ;\ \} \!\, floor floor; ⌊x⌋ means the floor of x, i.e. the largest integer less than or greatest equal to x. \lfloor ⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3 ⌊...⌋ \ldots integer; \rfloor \!\, entier (This may also be written [x], floor(x) or int(x).) numbers ceiling ⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x. \lceil ceiling ⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2 ⌈...⌉ \ldots \rceil \!\, numbers (This may also be written ceil(x) or ceiling(x).) nearest integer function ⌊x⌉ means the nearest integer tox . \lfloor ⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊−3.4⌉ = −3, ⌊4.49⌉ = 4 ⌊...⌉ nearest \ldots (This may also be written [x], ||x||, nint(x) or Round(x).) \rceil \!\, integer to numbers [ : ] degree of a [K : F] means the degree of the extensionK : F. [ℚ(√2) : ℚ] = 2 [\ :\ ] \!\, field extension [ℂ : ℝ] = 2 the degree of [ℝ : ℚ] = ∞ field theory equivalence class [a] means the equivalence class ofa , i.e. {x : x ~ a}, where ~ the Let a ~ b be true iff a ≡ b (mod 5). is an equivalence relation. equivalence Then [2] = {..., −8, −3, 2, 7, ...}. class of [a]R means the same, but withR as the equivalence relation. abstract algebra floor [x] means the floor of x, i.e. the largest integer less than or floor; equal to x. greatest [3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4 integer; (This may also be written ⌊x⌋, floor(x) or int(x). Not to be entier confused with the nearest integer function, as described numbers below.) nearest integer [x] means the nearest integer tox . function [2] = 2, [2.6] = 3, [−3.4] = −3, [4.49] = 4 nearest (This may also be written ⌊x⌉, ||x||, nint(x) or Round(x). Not to integer to be confused with the floor function, as described above.) numbers Iverson bracket 1 if true, 0 [ ] [S] maps a true statement S to 1 and a false statementS to 0. [0=5]=0, [7>0]=1, [2 ∈ {2,3,4}]=1, [5 ∈ {2,3,4}]=0 [\ ] \!\, otherwise propositional [ , ] logic [\ ,\ ] \!\, f[X] means { f(x) : x ∈ X }, the image of the functionf under image the set X ⊆ dom(f). [ , , ] image of ... under ... (This may also be written as f(X) if there is no risk of everywhere confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) closed interval closed . 0 and 1/2 are in the interval [0,1]. interval order theory commutator the [g, h] = g−1h−1gh (or ghg−1h−1), if g, h ∈ G (a group). xy = x[x, y] (group theory). commutator of [a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra). [AB, C] = A[B, C] + [A, C]B (ring theory). group theory, ring theory triple scalar product the triple scalar [a, b, c] = a × b · c, the scalar product of a × b with c. [a, b, c] = [b, c, a] = [c, a, b]. product of vector calculus function application f(x) means the value of the functionf at the element x. If f(x) := x2 − 5, then f(6) = 62 − 5 = 36 − 5=31. of set theory f(X) means { f(x) : x ∈ X }, the image of the functionf under image the set X ⊆ dom(f). image of ... under ... (This may also be written as f[X] if there is a risk of confusing everywhere the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. parentheses everywhere An ordered list (or sequence, or horizontal vector, or row ( ) tuple (\ ) \!\, vector) of values. (a, b) is an ordered pair (or 2-tuple). tuple; n-tuple; (Note that the notation (a,b) is ambiguous: it could be an ordered (a, b, c) is an ordered triple (or 3-tuple). ( , ) (\ ,\ ) \!\, pair/triple/etc; ordered pair or an open interval. Set theorists and computer row vector; scientists often use angle brackets ⟨ ⟩ instead of ( ) is the empty tuple (or 0-tuple). sequence parentheses.) everywhere

highest common factor highest common (a, b) means the highest common factor ofa and b. factor; (3, 7) = 1 (they are coprime); (15, 25) = 5. greatest (This may also be written hcf(a, b) or gcd(a, b).) common divisor; hcf; gcd number theory ( , ) open interval . 4 is not in the interval (4, 18). (\ ,\ ) \!\, open interval (Note that the notation (a,b) is ambiguous: it could be an (0, +∞) equals the set of positive real numbers. (\ ,\ ) \!\, order theory ] , [ ordered pair or an open interval. The notation ]a,b[ can be used instead.) ]\ ,\ [ \!\,] left-open interval ( , ] (\ ,\ ] \!\, half-open interval; . (−1, 7] and (−∞, −1] left-open ] , ] \ ,\ ] \!\,] interval order theory right-open interval [ , ) [\ ,\ ) \!\, half-open interval; . [4, 18) and [1, +∞) right-open [ , [ [\ ,\ [ \!\, interval order theory ⟨u,v⟩ means the inner product ofu and v, where u and v are members of an inner product space.

Note that the notation ⟨u, v⟩ may be ambiguous: it could mean inner product the inner product or the linear span. The standard inner product between two vectors x = (2, 3) and y = (−1, 5) inner product is: of There are many variants of the notation, such as ⟨u | v⟩ and (u ⟨x, y⟩ = 2 × −1 + 3 × 5 = 13 linear algebra | v), which are described below. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts. for a time series :g(t) (t = 1, 2,...) average let S be a subset of N for example, represents the we can define the structure functions Sq( ): average of average of all the elements in S. statistics

For a single discrete variable of a function , the expectation expectation value of is defined as value , and for a single continuous variable the expectation the expectation value of is defined as value of \langle\ probability ; where is the PDF of the variable ⟨⟩ \rangle \!\, theory .[16]

⟨,⟩ \langle\ ,\ ⟨S⟩ means the span of S ⊆ V. That is, it is the intersection of \rangle \!\, all subspaces of V which contain S. ⟨u1, u2, ...⟩ is shorthand for ⟨{u1, u2, ...}⟩. linear span (linear) span of; Note that the notation ⟨u, v⟩ may be ambiguous: it could mean . linear hull of the inner product or the linear span. linear algebra The span of S may also be written as Sp(S).

subgroup generated by a set means the smallest subgroup ofG (where S ⊆ G, a group) containing every element ofS . In S , and . the subgroup 3 is shorthand for . generated by group theory tuple is an ordered pair (or 2-tuple). tuple; n-tuple; An ordered list (or sequence, or horizontal vector, or row ordered vector) of values. is an ordered triple (or 3-tuple). pair/triple/etc; (The notation (a,b) is often used as well.) row vector; is the empty tuple (or 0-tuple). sequence everywhere ⟨u | v⟩ means the inner product ofu and v, where u and v are members of an inner product space.[17] (u | v) means the same. ⟨|⟩ \langle\ |\ inner product \rangle \!\, inner product Another variant of the notation is ⟨u, v⟩ which is described of above. For spatial vectors, thedot product notation, x · y is (|) linear algebra common. For matrices, the colon notation A : B may be used. (\ |\ ) \!\, As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts.

Other non-letter symbols Name Symbol Symbol Read as Explanation Examples in HTML in TeX Category convolution convolution; convolved with f ∗ g means the convolution off and g. . functional analysis complex conjugate z∗ means the complex conjugate ofz . conjugate . ( can also be used for the conjugate of z, as complex described below.) numbers

group of units R∗ consists of the set of units of the ringR , along with the operation of multiplication. the group of units of This may also be written R× as described ring theory above, or U(R). * hyperreal numbers the (set of) ∗R means the set of hyperreal numbers. ∗N is the hypernatural numbers. hyperreals Other sets can be used in place ofR . non-standard analysis Hodge dual ∗v means the Hodge dual of a vectorv . If v Hodge dual; is a k-vector within an n-dimensional oriented If are the standard basis vectors of , Hodge star inner product space, then ∗v is an (n−k)- linear algebra vector. Kleene star Corresponds to the usage of * inregular Kleene star expressions. If ∑ is a set of strings, then ∑* is If ∑ = ('a', 'b', 'c') then ∑* includes '', 'a', 'ab', 'aba', 'abac', computer the set of all strings that can be created by etc. The full set cannot be enumerated here since it is science, concatenating members of ∑. The same countably infinite, but each individual string must have mathematical string can be used multiple times, and the finite length. logic empty string is also a member of ∑*. proportionality is proportional to; y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. varies as everywhere Karp [18] reduction ∝ \propto \!\, is Karp reducible to; A ∝ B means the problem A can be is polynomial- If L ∝ L and L ∈ P, then L ∈ P. time many-one polynomially reduced to the problemB . 1 2 2 1 reducible to computational complexity theory set-theoretic complement A ∖ B means the set that contains all those minus; [13] elements of A that are not in B. without; {1,2,3,4} ∖ {3,4,5,6} = {1,2} ∖ \setminus throw out; (− can also be used for set-theoretic not complement as described above.) set theory conditional event P(A|B) means the probability of the eventA if X is a uniformly random day of the year P(X is May 25 | given occurring given that B occurs. X is in May) = 1/31 probability restriction restriction of ... f| means the function f is restricted to the A The function f : R → R defined by f(x) = x2 is not injective, to ...; set A, that is, it is the function withdomain A | but f| + is injective. restricted to ∩ dom(f) that agrees with f. R set theory such that S = {(x,y) | 0 < y < f(x)} such that; | means "such that", see ":" (described The set of (x,y) such that y is greater than 0 and less so that below). than f(x). everywhere a ∣ b means a divides b. a ∤ b means a does not divide b. ∣ \mid divisor, divides divides (The symbol ∣ can be difficult to type, and its Since 15 = 3 × 5, it is true that 3∣ 15 and 5 ∣ 15. number theory negation is rare, so a regular but slightly ∤ \nmid shorter vertical bar | character is often used instead.) exact divisibility pa ∣∣ n means pa exactly divides n (i.e. pa 23 ∣∣ 360. ∣∣ \mid\mid exactly divides divides n but pa+1 does not). number theory ∥ parallel x ∥ y means x is parallel to y. If l ∥ m and m ⊥ n then l ⊥ n. \| is parallel to x ∦ y means x is not parallel to y. x ⋕ y means x is equal and parallel to y. Requires the viewer to support geometry Unicode: \unicode{x2225}, ∦ \unicode{x2226}, and (The symbol ∥ can be difficult to type, and its \unicode{x22D5}. negation is rare, so two regular but slightly \mathrel{\rlap{\,\parallel}} longer vertical bar || characters are often ⋕ used instead.) requires incomparability \setmathfont{MathJax}.[19] is incomparable x ∥ y means x is incomparable to y. {1,2} ∥ {2,3} under set containment. to order theory cardinality #X means the cardinality of the setX . cardinality of; size of; #{4, 6, 8} = 3 (|...| may be used instead as described order of above.) set theory connected sum connected A#B is the connected sum of the manifoldsA sum of; # \sharp and B. If A and B are knots, then this denotes A#Sm is homeomorphic to A, for any manifold A, and the knot sum of; the knot sum, which has a slightly stronger m knot sphere S . condition. composition of topology, knot theory primorial n# is product of all prime numbers less than primorial 12# = 2 × 3 × 5 × 7 × 11 = 2310 or equal to n. number theory such that : means "such that", and is used in proofs such that; and the set-builder notation (described ∃ n ∈ ℕ: n is even. so that below). everywhere field extension K : F means the field K extends the field F. extends; ℝ : ℚ over This may also be written as K ≥ F. field theory A : B means the Frobenius inner product of inner product the matrices A and B. of matrices inner product The general inner product is denoted by ⟨u, : of v⟩, ⟨u | v⟩ or (u | v), as described below. For linear algebra spatial vectors, the dot product notation, x·y is common. See also bra–ket notation. index of a subgroup The index of a subgroup H in a group G is index of the "relative size" of H in G: equivalently, the subgroup number of "copies" (cosets) of H that fill up G group theory division divided by A : B means the division of A with B (dividing 10 : 2 = 5 over A by B) everywhere vertical ellipsis Denotes that certain constants and terms are ⋮ vertical ellipsis missing out (e.g. for clarity) and that only the \vdots \!\, everywhere important terms are being listed. wreath product A ≀ H means the wreath product of the group wreath product A by the group H. is isomorphic to the automorphism group of the ≀ \wr \!\, of ... by ... complete bipartite graph on (n,n) vertices. group theory This may also be written A wr H. \blitza \lighting: requires \usepackage{stmaryd}.[20] \smashtimes requires \usepackage{unicode-math} and \setmathfont{XITS Math} or another Open Type Math Font.[21] x + 4 = x − 3 ※ downwards [2] ↯ zigzag arrow Statement: Every finite, non-empty, ordered set has a \Rightarrow\Leftarrow Denotes that contradictory statements have largest element. Otherwise, let's assume that is a contradiction; been inferred. For clarity, the exact point of finite, non-empty, ordered set with no largestelement. this contradicts contradiction can be appended. Then, for some , there exists an with [2] that ⇒⇐ , but then there's also an with , \bot everywhere and so on. Thus, are distinct elements in . ↯ is finite. [2] \nleftrightarrow

\textreferencemark[2] Contradiction![2]

⊕ exclusive or \oplus \!\, xor The statement A ⊕ B is true when either A or

propositional B, but not both, are true. A ⊻ B means the (¬A) ⊕ A is always true, A ⊕ A is always false.

logic, Boolean same. ⊻ \veebar \!\, algebra direct sum The direct sum is a special way of combining Most commonly, for vector spaces U, V, and W, the direct sum of several objects into one general object. following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) abstract (The bun symbol ⊕, or the coproduct symbol algebra ∐, is used; ⊻ is only for logic.) Kulkarni– Nomizu Derived from the tensor product of two product symmetric type (0,2) tensors; it has the algebraic symmetries of theRiemann tensor. Kulkarni– {~\wedge\!\!\!\!\!\!\bigcirc~} has components Nomizu product . tensor algebra D'Alembertian; It is the generalisation of theLaplace wave operator operator in the sense that it is the differential operator which is invariant under the non-Euclidean □ \Box \!\ isometry group of the underlying space and it Laplacian reduces to the Laplace operator if restricted vector calculus to time independent functions.

Letter-based symbols

Includes upside-down letters.

Letter modifiers Also called diacritics.

Name Symbol Symbol Read as Explanation Examples in HTML in TeX Category mean (often read as "x bar") is themean (average overbar; . ... bar value of ). statistics finite sequence, tuple means the finite sequence/tuple finite . sequence, . tuple model theory algebraic closure a \bar{a} The field of algebraic numbers is sometimes denoted as algebraic is the algebraic closure of the fieldF . closure of because it is the algebraic closure of therational numbers . field theory complex conjugate means the complex conjugate ofz . conjugate (z∗ can also be used for the conjugate of z, as . complex described above.) numbers topological closure is the topological closure of the setS . (topological) In the space of the real numbers, (the rational numbers are dense in the real numbers). closure of This may also be denoted as cl(S) or Cl(S). topology vector

harpoon \overset{\rightharpoonup} {a} linear algebra unit vector (pronounced "a hat") is thenormalized version hat of vector , having length 1. geometry

â \hat a estimator estimator is the estimator or the estimate for the The estimator produces a sample estimate for for parameter . the mean . statistics derivative f ′(x) means the derivative of the functionf at the point x, i.e., the slope of the tangent to f at x. ... prime; If f(x) := x2, then f ′(x) = 2x. ′ ' derivative of (The single-quote character ' is sometimes used calculus instead, especially in ASCII text.) derivative ... dot; means the derivative of x with respect to time.

• time If x(t) := t2, then . \dot{\,} That is . derivative of calculus

Symbols based on Latin letters Name Read as Symbol Symbol Explanation Examples in HTML in TeX Category

universal quantification for all; for any; means P(x) is true for all x. 2 ∀ \forall for each; ∀ x, P(x) ∀ n ∈ ℕ, n ≥ n. for every predicate logic boolean domain 픹 \mathbb{B} B; the (set of) boolean values; 픹 means either {0, 1}, {false, true}, {F, T}, or . (¬False) ∈ 픹 B the (set of) truth values; \mathbf{B} set theory, boolean algebra complex numbers ℂ \mathbb{C} C; ℂ means . the (set of) complex numbers {a + b i : a,b ∈ ℝ} i = √−1 ∈ ℂ

C \mathbf{C} numbers cardinality of the continuum cardinality of the continuum; The cardinality of is denoted by or by the symbol (a c; \mathfrak c lowercase Fraktur letter C). cardinality of the real numbers set theory partial derivative partial; ∂f/∂x means the partial derivative off with respect to x , where f is i i If f(x,y) := x2y, then ∂f/∂x = 2xy, d a function on (x1, ..., xn). calculus boundary ∂ \partial boundary of ∂M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} topology degree of a polynomial ∂f means the degree of the polynomialf . degree of ∂(x2 − 1) = 2 algebra (This may also be written deg f.)

피 \mathbb E expected value the value of a random variable one would "expect" to find if one expected value could repeat the random variable process an infinite number of probability theory times and take the average of the values obtained E \mathrm{E} existential quantification there exists; ∃ \exists there is; ∃ x: P(x) means there is at least onex such that P(x) is true. ∃ n ∈ ℕ: n is even.
there are predicate logic uniqueness quantification

there exists exactly one ∃! x: P(x) means there is exactly onex such that P(x) is true. ∃! n ∈ ℕ: n + 5 = 2n. ∃! \exists! predicate logic set membership ∈ \in (1/2)−1 ∈ ℕ is an element of; a ∈ S means a is an element of the set S;[15] a ∉ S means a is is not an element of not an element of S.[15] 2−1 ∉ ℕ ∉ \notin everywhere, set theory set membership S ∌ e means the same thing ase ∉ S, where S is a set and e is does not contain as an element ∌ \not\ni not an element of S. set theory often abbreviated as "s.t."; : and | are also used to abbreviate such that symbol "such that". The use of ∋ goes back to early mathematical logic Choose ∋ 2| and 3| . (Here | is such that and its usage in this sense is declining. The symbol ("back used in the sense of "divides".) epsilon") is sometimes specifically used for "such that" to avoid mathematical logic confusion with set membership. ∋ \ni set membership S ∋ e means the same thing ase ∈ S, where S is a set and e is contains as an element an element of S. set theory quaternions or Hamiltonian ℍ \mathbb{H} quaternions H; ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}. H the (set of) quaternions \mathbf{H} numbers N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.

The choice depends on the area of mathematics being studied;

e.g. number theorists prefer the latter; analysts, set theorists and ℕ \mathbb{N} natural numbers computer scientists prefer the former. To avoid confusion, always ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a the (set of) natural numbers check an author's definition of N. ∈ ℤ} numbers N \mathbf{N} Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤.

Hadamard product For two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same entrywise product ○ \circ dimensions with elements given by linear algebra . ∘ function composition f ∘ g is the function such that (f ∘ g)(x) = f(g(x)).[22] if f(x) := 2x, and g(x) := x + 3, then (f \circ composed with ∘ g)(x) = 2(x + 3). set theory Big O notation The Big O notation describes the limiting behavior of a function, If f(x) = 6x4 − 2x3 + 5 and g(x) = x4, big-oh of O O when the argument tends towards a particular value orinfinity . then Computational complexity theory

∅ \empty empty set the empty set null set ∅ means the set with no elements.[15] { } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅ \varnothing set theory { } \{\} set of primes P; ℙ is often used to denote the set of prime numbers. the set of prime numbers arithmetic projective space P; the projective space; ℙ means a space with a point at infinity. , the projective line; ℙ \mathbb{P} the projective plane topology

probability ℙ(X) means the probability of the eventX occurring. P \mathbf{P} If a fair coin is flipped, ℙ(Heads) = the probability of ℙ(Tails) = 0.5. probability theory This may also be written as P(X), Pr(X), P[X] or Pr[X]. The power set P({0, 1, 2}) is the set Given a set S, the power set of S is the set of all subsets of the set Power set of all subsets of {0, 1, 2}. Hence, S. The power set of S0 is the Power set of P({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, denoted by P(S). Powerset {0, 2}, {1, 2}, {0, 1, 2} }.

rational numbers 3.14000... ∈ ℚ ℚ \mathbb{Q} Q; the (set of) rational numbers; ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. π ∉ ℚ Q the rationals \mathbf{Q} numbers

real numbers π ∈ ℝ ℝ \mathbb{R} R; the (set of) real numbers; ℝ means the set of real numbers. √(−1) ∉ ℝ R the reals \mathbf{R} numbers conjugate transpose conjugate transpose; A† means the transpose of the complex conjugate ofA .[23] † adjoint; † If A = (aij) then A = (aji). {}^\dagger Hermitian ∗T T∗ ∗ T T adjoint/conjugate/transpose/dagger This may also be written A , A , A , A or A . matrix operations transpose AT means A, but with its rows swapped for columns. T T transpose If A = (aij) then A = (aji). {}^{\mathsf{T}} t tr matrix operations This may also be written A′, A or A . top element the top element ⊤ means the largest element of a lattice. ∀x : x ∨ ⊤ = ⊤ lattice theory ⊤ \top top type ⊤ means the top or universal type; every type in thetype system the top type; top ∀ types T, T <: ⊤ of interest is a subtype of top. type theory perpendicular x ⊥ y means x is perpendicular to y; or more generally x is If l ⊥ m and m ⊥ n in the plane, is perpendicular to orthogonal to y. then l || n. geometry orthogonal complement orthogonal/ perpendicular W⊥ means the orthogonal complement ofW (where W is a complement of; subspace of the inner product space V), the set of all vectors in V Within , . perp orthogonal to every vector inW . linear algebra coprime is coprime to x ⊥ y means x has no factor greater than 1 in common withy . 34 ⊥ 55 number theory independent A ⊥ B means A is an event whose probability is independent of event B. The double perpendicular symbol ( ) is also commonly is independent of If A ⊥ B, then P(A|B) = P(A). ⊥ \bot used for the purpose of denoting this, for instance: (In probability LaTeX, the command is: "A \perp\!\!\!\perp B".) bottom element the bottom element ⊥ means the smallest element of a lattice. ∀x : x ∧ ⊥ = ⊥ lattice theory bottom type the bottom type; ⊥ means the bottom type (a.k.a. the zero type or empty type); ∀ types T, ⊥ <: T bot bottom is the subtype of every type in thetype system. type theory {e, π} ⊥ {1, 2, e, 3, π} under set comparability containment. is comparable to x ⊥ y means that x is comparable to y. order theory

all numbers being considered 핌 means "the set of all elements being considered." 핌 = {ℝ,ℂ} includes all numbers. 핌 \mathbb{U} It may represent all numbers both real and complex, or any subset U; of these—hence the term "universal". If instead, 핌 = {ℤ,ℂ}, then π ∉ 핌. the universal set; U \mathbf{U} the set of all numbers; all numbers considered set theory set-theoretic union the union of ... or ...; A ∪ B means the set of those elements which are either inA , or in A ⊆ B ⇔ (A ∪ B) = B ∪ \cup union B, or in both.[13] set theory set-theoretic intersection intersected with; A ∩ B means the set that contains all those elements thatA and B {x ∈ ℝ : x2 = 1} ∩ ℕ = {1} ∩ \cap intersect have in common.[13] set theory logical disjunction or join in a lattice The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. or; n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a ∨ \lor max; natural number. For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), join B(x)). propositional logic, lattice theory logical conjunction or meet in a lattice The statement A ∧ B is true if A and B are both true; else it is false. and; n < 4 ∧ n > 2 ⇔ n = 3 when n is a min; natural number. For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), meet B(x)). ∧ \land propositional logic, lattice theory wedge product u ∧ v means the wedge product of anymultivectors u and v. In wedge product; three-dimensional Euclidean space the wedge product and the exterior product cross product of two vectors are each other's Hodge dual. exterior algebra multiplication 3 × 4 means the multiplication of 3 by 4. times; 7 × 8 = 56 multiplied by (The symbol * is generally used in programming languages, where arithmetic ease of typing and use ofASCII text is preferred.) Cartesian product X × Y means the set of all ordered pairs with the first element of the Cartesian product of ... and ...; {1,2} × {3,4} = {(1,3),(1,4),(2,3), each pair selected from X and the second element selected from the direct product of ... and ... (2,4)} Y. set theory × \times cross product (1,2,5) × (3,4,−1) = cross u × v means the cross product ofvectors u and v (−22, 16, − 2) linear algebra

group of units R× consists of the set of units of the ringR , along with the operation of multiplication. the group of units of ring theory This may also be written R∗ as described below, or U(R). tensor product, tensor product of {1, 2, 3, 4} ⊗ {1, 1, 2} = modules means the tensor product ofV and U.[24] means {{1, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, ⊗ \otimes tensor product of the tensor product of modulesV and U over the ring R. 8}} linear algebra

N ⋊φ H is the semidirect product ofN (a normal subgroup) andH semidirect product (a subgroup), with respect to φ. Also, ifG = N ⋊φ H, then G is said ⋉ the semidirect product of to split over N. \ltimes group theory (⋊ may also be written the other way round, as ⋉, or as ×.)

⋊ \rtimes semijoin R ⋉ S is the semijoin of the relationsR and S, the set of all tuples R S = (R S) the semijoin of in R for which there is a tuple inS that is equal on their common a1,..,an relational algebra attribute names. natural join R ⋈ S is the natural join of the relationsR and S, the set of all

the natural join of combinations of tuples inR and S that are equal on their common ⋈ \bowtie relational algebra attribute names. ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}. ℤ+ or ℤ> means {1, 2, 3, ...} . ℤ \mathbb{Z} integers ℤ≥ means {0, 1, 2, 3, ...} . the (set of) integers ℤ = {p, −p : p ∈ ℕ ∪ {0}} * [25] Z numbers ℤ is used by some authors to mean {0, 1, 2, 3, ...} and others \mathbf{Z} to mean {... -2, -1, 1, 2, 3, ... }[26] .

ℤn ℤn means {[0], [1], [2], ...[n−1]} with addition and multiplication \mathbb{Z}_n integers mod n modulo n. the (set of) integers modulon ℤ3 = {[0], [1], [2]} ℤp numbers Note that any letter may be used instead of n, such as p. To avoid \mathbb{Z}_p confusion with p-adic numbers, use ℤ/pℤ or ℤ/(p) instead. Z n \mathbf{Z}_n p-adic integers the (set of) p-adic integers Note that any letter may be used instead of p, such as n or l. Zp numbers

Symbols based on Hebrew or Greek letters Name Symbol Symbol Read as Explanation Examples in HTML in TeX Category aleph number ℵ represents an infinite cardinality (specifically, the α-th α |ℕ| = ℵ , which is called aleph-null. ℵ \aleph aleph one, where α is an ordinal). 0 set theory beth number ℶ represents an infinite cardinality (similar toℵ , but ℶ does beth α ℶ \beth not necessarily index all of the numbers indexed byℵ . ). set theory Dirac delta function δ(x) Dirac delta of hyperfunction Kronecker delta

Kronecker δij delta of δ \delta hyperfunction

Functional derivative Functional derivative of Differential operators

∆ \vartriangle symmetric A ∆ B (or A ⊖ B) means the set of elements in exactly one of difference {1,5,6,8} ∆ {2,5,8} = {1,2,6} A or B. symmetric ⊖ \ominus difference {3,4,5,6} ⊖ {1,2,5,6} = {1,2,3,4} (Not to be confused with delta, Δ, described below.) set theory ⊕ \oplus

delta Δx means a (non-infinitesimal) change inx . delta; (If the change becomes infinitesimal, δ and even d are used is the gradient of a straight line. change in instead. Not to be confused with the symmetric difference, calculus written ∆, above.)

Δ \Delta Laplacian Laplace The Laplace operator is a second order differential operator If ƒ is a twice-differentiable real-valued function, then the Laplacian operator in n-dimensional Euclidean space of ƒ is defined by vector calculus gradient del; nabla; ∇f (x , ..., x ) is the vector of partial derivatives ∂f( / ∂x , ..., 1 n 1 If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) gradient of ∂f / ∂xn). vector calculus divergence del dot; ∇ \nabla divergence of If , then . vector calculus

curl

curl of If , then . vector calculus

Pi Used in various formulas involving circles; is equivalent to pi; π the amount of area a circle would take up in a square of 3.1415926...; equal width with an area of 4 square units, roughly 3.14159. A = πR2 = 314.16 → R = 10 ≈355÷113 It is also the ratio of the circumference to the diameter of a mathematical circle. constant projection Projection of restricts to the attribute set. π \pi relational algebra Homotopy group the nth consists of homotopy equivalence classes of base Homotopy point preserving maps from an n-dimensional sphere (with group of base point) into the pointed space X. Homotopy theory product ∏ product over \prod ... from ... to means . ... of arithmetic Cartesian product means the set of all (n+1)-tuples the Cartesian product of; (y0, ..., yn). the direct product of set theory coproduct A general construction which subsumes thedisjoint union of coproduct sets and of topological spaces, the free product of groups, over ... from and the direct sum of modules and vector spaces. The ∐ \coprod ... to ... of coproduct of a family of objects is essentially the "least category specific" object to which each object in the family admits a theory morphism. selection The selection selects all those tuples in for which σ Selection of holds between the and the attribute. The selection \sigma relational selects all those tuples in for which holds algebra between the attribute and the value . summation sum over ...

∑ from ... to ... means . \sum of arithmetic

Variations

In mathematics written in Persian orArabic , some symbols may be reversed to make right-to-left writing and reading easier.[27]